Persistent Homology Computation Using Combinatorial Map Simplification

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Persistent Homology Computation Using Combinatorial

Map Simplification

Guillaume Damiand, Rocio Gonzalez-Diaz

To cite this version:

Combinatorial Map Simplification

Guillaume Damiand and Rocio Gonzalez-Diaz

1

Univ. Lyon, CNRS, LIRIS, UMR5205, F-69622 France guillaume.damiand@liris.cnrs.fr

https://liris.cnrs.fr/guillaume.damiand/

2

Universidad de Sevilla, Dpto. de Matem´atica Aplicada I, S-41012, Spain rogodi@us.es

http://personal.us.es/rogodi/

Abstract. We propose an algorithm for persistence homology computa-tion of orientable 2-dimensional (2D) manifolds with or without bound-ary (meshes) represented by 2D combinatorial maps. Having as an input a real function h on the vertices of the mesh, we first compute persis-tent homology of filtrations obtained by adding cells incident to each vertex of the mesh, The cells to add are controlled by both the function h and a parameter δ. The parameter δ is used to control the number of cells added to each level of the filtration. Bigger δ produces less levels in the filtration and consequently more cells in each level. We then simplify each level (cluster) by merging faces of the same cluster. Our experiments demonstrate that our method allows fast computation of persistent ho-mology of big meshes and it is persistent-hoho-mology aware in the sense that persistent homology does not change in the simplification process when fixing δ.

Keywords: Persistent homology computation; 2D combinatorial map; mesh simplification

1

Introduction

Topological data analysis (TDA) is a relatively new field in computer science. One of the most useful concept in TDA is the one of persistent homology which is an algebraic method for measuring topological features (connected components, voids, cavities, etc) of shapes and functions. Two of the crucial ingredients of persistence are: (1) a cell complex to structure the data; and (2) a filtration which is a nested sequence of subcomplexes that starts with the empty complex and ends with the whole complex. See [1, 2] for initial reports and [3, 4] for a modern exposition of the field.

(GVF) by a process presented in [6]. To do this, the cell complex is decomposed into the lower star of its vertices. The authors then computed persistent homol-ogy from the boundary map of the chain complex associated to the Morse-Smale complex induced by GVF.

In [7], we proposed an efficient algorithm for computing the homology of meshes (orientable manifolds with or without boundary), represented by 2D combinatorial maps (which are models of representation of meshes composed by vertices, edges linking two vertices, and 2D faces bounded by a closed path of edges), avoiding the time-consuming step of constructing and modifying bound-aries and coboundbound-aries of cells. The process consists of merging faces if they share a common edge, guaranteeing that the structure of combinatorial map and the homology information of the mesh is preserved until the end of the process.

In this paper we extend our work to compute persistent homology of meshes. First, as in [7], a simplification process is made to improve computation time. Now, faces as dispatched in clusters depending on a parameter δ and only faces of the same cluster are merged. For constructing the cluster the following rule is used: two faces are in a same cluster if there is a path of vertices of these two faces of length smaller than δ. At the end of the process, a smaller than the input 2D combinatorial map is obtained. To obtain persistent homology of the simplified mesh, lower-start filtration induced by a function h on its vertices (in our case, h is the height function) is computed. Varying the parameter δ, the filtration varies and also its persistent homology.

The paper is organized as follows. Section 2 recalls the background of the pa-per regarding combinatorial maps and pa-persistent homology. Section 3 is the main section of the paper and presents our method to compute persistent homology starting from a particular filtration constructed from the height function and a parameter δ. Several experimental and computational results are presented in Section 4. Finally, we summarize the paper with a brief discussion about future work in Section 5.

2

Preliminary Notions

In this section we recall the needed background of the paper regarding combi-natorial maps and persistent homology.

2.1 2D combinatorial maps

A 2D combinatorial map [8, 9], called 2-map, is a model of representation of a mesh, which is composed by i-cells: vertices or 0-cells associated with points, edges or 1-cells which link two vertices, and faces or 2-cells which are bounded by a closed path.

An edge incident to two different faces is called inner. Such an edge is necessarily not dangling nor isolated. Lastly, an edge is called border if it is incident to only one face and if it touches the boundary of the mesh. See Figure 1(a).

f

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f

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e

f

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Fig. 1. (a) Example of a mesh having 5 faces (the four faces incident to vertex v1,

and the “degenerated one” bounded twice by edge e7), 14 edges (e6 is dangling, e7

is isolated, {e1, e2, e3, e4} are inner and the rest are border) and 12 vertices. (b) The

corresponding 2-map has 20 darts. Images taken from [7].

The different elements of a mesh are encoded in a 2-map by darts and two mappings between these dart: β1 and β2:

β2: A dart is an orientation of an edge. If an edge separates two faces, it is described by two darts d1, d2 in the 2-map linked by β2 (i.e., β2(d1) = d2 and β2(d2) = d1). These two darts represent the two possible orientations of the edge (for example β2(8) = 11 and β2(11) = 8 in Figure 1(b)). Each border edge is described by only one dart d in the 2-map, linked by β2with a special element ∅ (cf. for example dart 10 in Figure 1(b) which describes border edge e5).β1: For each dart d, β1(d) is the dart following dart d and belonging to the same face than d (for example β1(1) = 2 in Figure 1(b)). Note that a 2-map is oriented and thus described a given orientation of the mesh.

A dart belongs exactly to one vertex, one edge and one face, and thus each cell of the mesh is described by a set of darts in the 2-map. For example, in Figure 1(b), vertex v1 is described by the set of darts {2, 5, 8, 12}. Note that this is a very important property of 2-map. Even an isolated edge (like e7 in Figure 1(a)) belongs to one face (which explain why we have 5 faces and not 4 in Figure 1(a)).

The different type of edges can be detected in a 2-map thanks to particular configurations of darts and β links (for example an edge is isolated if β1(β1(d)) = d, d being of of the dart of the edge).

cells) describing a given mesh. The algorithm uses two operations on 2-maps: edge removal and edge contraction. It simplifies a given combinatorial map in its minimal form while preserving all the homology information. The proof that Algorithm 1 preserves homology information is given in [10].

Algorithm 1: Simplification of a mesh (modified version of Algorithm 1 of [7]).

Input: A 2-map M representing the mesh.

Output: The simplified 2-map corresponding to M. foreach edge e of M do

if e is an inner edge then remove e; foreach edge e of M do

while e is dangling do

e0← one edge adjacent to e; remove e; e ← e0;

foreach edge e of M do if e is not a loop then

contract e;

2.2 Persistent Homology

In this subsection we give elementary notions from topology needed to under-stand the rest of the paper. In particular, we introduce the notion of homology and persistent homology. Precise definitions of homology can be found for ex-ample in [11], and definition of persistent homology for exex-ample in [4].

Homology can be thought as a method for defining k-dimensional holes (con-nected components, tunnels, voids) in a given mesh. For example. a 1-cycle is a closed path and a boundary is the boundary of a 2D manifold. Then, 1-homology classes (which represent tunnels) are equivalence classes of 1-cycles modulo 1-boundaries. This concept can be generalized to k-homology classes. Finally, k-homology groups are the groups of k-homology classes.

Persistent homology captures the topological changes occurring in a growing sequence of meshes, called filtration. During the growth of a mesh, homology classes of different dimension may appear (be born) and disappear (die). Filtra-tions are frequently constructed using a real-valued function h on the vertices of the mesh M . For example, the lower-start filtration is computed as follows:

– First, order the vertices of M in a non-decreasing way, h(v1) ≤ h(v2) ≤ · · · ≤ h(vn).

– Define Mi as the union of the lower-star of all vertices of M whose function value is at most h(vi).

This way, if h(vi−1) < h(vi) then Mi\ Mi−1is the set of cells of M having a vertex with function value exactly h(vi).

And if h(vi−1) = h(vi) then Mi−1= Mi.

The lower-star filtration of the mesh M induced by the function h is the sequence of nested meshes:

∅ = M0⊆ M1⊆ M2⊆ · · · ⊆ Mn−1⊆ Mn= M.

Intuitively, imagine we sweep the mesh M in increasing values of the function h. At any real-value α, we consider the set of cells whose function value on their vertices is below or equal to α. As α increases, this gives us a sequence of subsets of M , growing larger and larger.

The topological evolution along the filtration is expressed by the correspond-ing sequence of homology groups. When addcorrespond-ing the cells in order accordcorrespond-ing to the filtration, new homology classes may born and some of them may later die when they become trivial or merge with another class. If a homology class γ is born at Mi and dies entering Mj then h(vj) − h(vi) is the persistence of γ. If γ is born at Mi but never dies then its persistence is set to infinity. Homology classes with low persistence are considered noise and the ones that persist are considered features of the mesh.

The information obtained when computing persistent homology can be visu-alized as a persistence barcode which consists of the set of (birth, death) intervals, each interval recording a persistent homology event. The bottleneck distance is used to compare two persistence barcodes corresponding to two different filtra-tions of the same mesh. Given a bijection η between two persistence barcodes, we take the supremum L∞-distance3 between matched points and define the bottleneck distance by taking the infimum over all supremums.

In order to compute persistent homology, in this paper we have implemented a simplified version of the incremental algorithm for computing AT-models given in [12]. Given an ordering of the cells of the mesh, Algorithm 2 computes a triplet (M, H, f ) where:

– M is the given mesh (decomposed in cells obtained from the combinatorial map). If σ is a k-cell, then ∂(σ) is the set of (k − 1)-cells in its boundary. – H is a subset of cells of M called surviving cells. Fixed k, the set of all

the surviving k-cells together with the addition operation + (here + means the disjoint union of sets) form the group Ck(H) which is isomorphic to the k-dimensional homology group Hk of M .

– f : C(M ) → C(H) maps each k-cell in M to a sum of surviving cells, satisfying that if a, b ∈ Ck(M ) are two homologous k-cycles then fk(a) = fk(b). Let Mσi be the set of cells {σ1, . . . , σi}. Then, in the ith step of

3

The L∞-distance between points u = (u1, u2) and v = (v1, v2) in the extended plane

Algorithm 2: Computing persistent homology (Algorithm 2 of [12]).

Input: An ordering of the cells of M : {σ1, . . . , σm}.

Output: Persistent homology.

Initialize H := ∅ and f (σi) := 0, for 1 ≤ i ≤ m.

for i = 1 to m do if f ∂(σi) = 0 then

f (σi) := σi, H := H ∪ {σi} (a new homology class was born).

if f ∂(σi) 6= 0 then

Let σj∈ f ∂(σi) s t. j = max{ index(µ) : µ ∈ f ∂(σi) }

H := H \ {σj} (an old homology class died).

foreach x ∈ M such that σj∈ f (x) do

f (x) := f (x) + f ∂(σi).

Algorithm 2, σi belongs to a k-cycle c in C(Mσi) if and only if f ∂(σi) = 0.

This is why if f ∂(σi) = 0 then a new homology class was born (the one represented by the k-cycle c) and σi enters H. Otherwise, if f ∂(σi) 6= 0, then a homology class died, which is equivalent to say that an element of f ∂(σ) ⊆ H is removed from H. The element to be removed from H will be the youngest one: max{ index(µ) : µ ∈ f ∂(σi) }, being index(µ) the position of the cell µ in the given ordered list of cells {σ1, . . . , σm}.

In [13] the authors establish a correspondence between the incremental al-gorithm for computing AT-models given in [12] and the one for computing per-sistent homology [4]. Since we are only interested in computing the persistence events, we only compute the set H and the map f . See Algorithm 2.

3

Computing Persistence

Our starting point is a subdivision of a mesh M (with or without boundaries) into vertices, edges and faces, and a real-valued function h on the vertices of the mesh.

Our method is based on three steps:

1. Simplification of the 2-map according to a parameter δ; 2. Computation of the lower-star filtration of the simplified mesh; 3. Computation of persistent homology of the given filtration.

3.1 2-Map Simplification

In this step, the 2-map is simplified by dispatching the faces into clusters and applying Algorithm 1 with constraints.

First, faces are dispatched into clusters according to the parameter δ. To compute such clusters, vertices of the mesh are ordered in a non-decreasing way by their height values h(v). We assign a height value to each face with is the maximum value of the height of its vertices.

Then in the first cluster we add the first face f in the ordering and all the faces “at distance” less than δ. which means that there exists a path of vertices of these two faces of length smaller than δ. For example, if δ = 0, only one face per cluster is added. If δ = 1 all the faces sharing an edge with f are added. For any δ > 1 all the faces at distance less than or equal to δ to f are added to the cluster. We repeat the process with the next face provided by the ordering that was not included in any cluster. We repeat the process until all faces are in a cluster.

After dispatching the faces in clusters, we apply Algorithm 1 with the fol-lowing constraints:

– Faces merge (i.e, the inner shared edge e is removed) only if they belong to the same cluster.

– Besides, contrary to Algorithm 1, critical edges (separating faces belonging to two different clusters) are not removed here. Merging faces belonging to two different clusters could lead to loose a persistent event, and this is why we do not merge such faces.

– We do not use the contraction step (last foreach in Algorithm 1). Indeed, the simplified 2-map obtained here has several faces, contrary to Algorithm 1 computed without constraints that always produces one face per connected component. For this reason, the number of possible edges to contract is here smaller and thus we have observed no gain (and even sometimes a loss) when using the contraction step comparing to not use it.

3.2 Filtration

The second step in our algorithm for computing persistent homology is to com-pute the lower-star filtration (see Section 2.2) of the simplified mesh SM .

3.3 Computation of persistent homology

The last step of our method is the computation of persistent homology of the simplied mesh SM .

We order the cells in SM according to the given filtration and obtain the ordered set of cells {σ1, . . . , σm} such that if i < j then there exist i0, j0 such that i0 ≤ j0_{, σ}

i∈ SMi0, σ_j∈ SM_j0 and σ_j is not in the boundary of σ_i. We then

apply Algorithm 2 to compute persistent homology.

The persistence barcode is stored in a list L with the (birth, death) events as follows: if σ ∈ M`\ M`−1 is born and dies entering µ ∈ Mm\ Mm−1, then store (birth, death) in L being birth= h(vi`) and death= h(vim).

Finally, bottleneck distance between different filtrations of the same mesh obtained from different values of δ can be computed to measure the effect of the parameter δ in the persistent homology information obtained.

4

Experiments

We have implemented our algorithm for persistent homology computation by using the CGAL implementation of combinatorial maps [14] and the additional layer, called linear cell complex, which additionally represents the geometry [15]. All our experiments were run on an Intel i7-4790 CPU, 4 cores @ 3.60GHz withR 32 Go RAM. All the computation times given here are averages of 10 consecutive runs of the same method.

In our tests, we used the six meshes shown in Figure 2, having between 703, 512 and 10, 000, 000 faces. All these meshes have only one connected com-ponent, except Blade which has 295 connected components because it contains many small isolated closed meshes inside the blade.

In our experiment, we compared the persistent homology computation of the six meshes for the following values of δ: 0, 1, 2, 4, 8, 16, 32 and 64. For δ = 0, the persistent homology computed is the one of the lower-star filtration induced by the height function on the vertices of the original mesh. When δ increases, the number of faces in a same cluster increases also and thus the combinatorial map becomes more and more simplified. Nevertheless, persistent homology varies since the filtration varies, although differences are “small”.

(a) (b)

(c) (d) (e) (f)

#0-cells #1-cells #2-cells #H0 #H1 #H2 (a) Blade 882,954 2,648,082 1,765,388 295 330 295 (b) DrumDancer 1,335,436 4,006,302 2,670,868 1 0 1 (c) Neptune 2,003,932 6,011,808 4,007,872 1 6 1 (d) HappyBuddha 543,652 1,631,574 1,087,716 1 208 1 (e) Iphigenia 351,750 1,055,268 703,512 1 8 1 (f) ThaiStatue 4,999,996 15,000,000 10,000,000 1 6 1

(a) (b) (c) (d)

Fig. 3. Effect of the δ parameter on the size of the different clusters for the Neptune mesh, zoom in on the trident. (a) δ = 0. (b) δ = 1. (c) δ = 4. (c) δ = 32.

4096 16,384 65,536 262,144 1,048,576 4,194,304 0 2 4 8 16 32 64 Number of cells Delta 0-cells 1-cells 2-cells

Fig. 4. Number of vertices, edges and faces of the simplified combinatorial maps (in log2 scale) depending on the value of δ. δ = 0 is the original (non-simplified) 2-map.

These numbers are average values for the six meshes.

δ 1 2 4 8 16 32 64 Blade 0.64 1 1.53 2.5 3.43 16.25 10.30 DrumDancer 0.10 0.87 0.62 1.25 1.18 3.31 3.31 Neptune 1.10 1.25 1.67 3.08 5.41 8.00 13.41 HappyBuddha 0.00025 0.0005 0.0014 0.0017 0.0024 0.0060 0.010 Iphigenia 0.88 1.19 1.64 2.71 4.51 9.87 19.22 Statuette 0.90 12.37 12.37 12.24 18.39 26.20 27.41

0 1 2 3 4 5 6 7 8 0 1 2 4 8 16 32 64 Time (sec) Delta Filtration Simplification AT-model δ 0 1 2 4 8 16 32 64 Blade 13.81 5.94 1.69 1.14 0.93 0.80 0.72 0.69 DrumDancer 2.73 2.37 1.79 1.62 1.48 1.36 1.24 1.19 Neptune 8.42 5.23 3.59 2.92 2.63 2.46 2.29 2.26 HappyBuddha 3.39 1.92 0.99 0.73 0.62 0.56 0.52 0.50 Iphigenia 0.77 0.58 0.40 0.35 0.34 0.31 0.29 0.28 Statuette 13.75 10.13 7.15 6.05 5.30 4.89 4.69 4.49 Average 7.15 4.36 2.60 2.14 1.88 1.73 1.63 1.57

Fig. 5. Computation time (in seconds) of our method by using the patch filtration with increasing δ starting from 0 and going to 64. The graph shows average values for the six meshes, and details time spent in the different parts of the method: computation of the filtration, combinatorial map simplification and persistence computation by using AT-model. The array gives global computation time for each mesh.

We can see in Figure 6 the effect of δ on the results of the persistent homology computation. First, it should be notice that infinite events are always the same whatever the value of δ is. This is a direct consequence of the fact that the homology of the mesh is preserved by our simplification algorithm. For finite events, we can see that their numbers decrease when δ increase. Indeed, the combinatorial map becomes more and more simplified, and thus the number of cells becomes smaller and smaller (as seen in Figure 4).

In Table 1 we can see the bottleneck distance with respect to the 0-dimensional persistent homology between the persitence barcodes corresponding to the lower-star filtration and the filtration obtained when varying δ. Table 2 shows the same information for the 1-dimensional persistent homology. To compute the bottle-neck distance we used the package TDA of R4_{. We can observe that, in general,} the distance increases when δ increases and the distance is bounded by the value

4

4 16 64 256 1024 4096 16,384 0 2 4 8 16 32 64 Number of events Delta Betti 0 Bettti 1

Fig. 6. Number of finite persistence events (in log₂ scale) depending on the value of δ. δ = 0 is the original (non-simplified) 2-map. Betti i is the number of i-homology classes that were born and later died when computing persistent homology, for i = 0, 1. These numbers are average values for the six meshes.

of δ. Sometimes, δ increases and the distance is a bit lower. This could occurs due to small pockets in the considered mesh. Moreover we can see that in some meshes the effect of δ is more important than in others. See for example Table 1: for Statuette, the difference between the bottleneck distance for δ = 0 and δ = 4 and for δ = 0 and δ = 8 is only 12.37 − 12.24 = 0.13 which means that we obtain similar persistent homology information when computing persistent homology using δ = 8 instead of δ = 4. Nevertheless, bottleneck distance for δ = 0 and δ = 8 and for δ = 0 and δ = 16 is 18.39 − 12.24 = 6.14 which means that we could loss important details if we simplify the mesh using δ = 16 instead that δ = 8. δ 1 2 4 8 16 32 64 Blade 0.97 1.0 2.0 4.0 7.0 14.0 21.0 DrumDancer 0.14 0.19 0.38 0.58 1.06 2.40 1.89 Neptune 0.53 1.32 1.82 3.15 5.15 8.28 12.41 HappyBuddha 0.00039 0.00067 0.0010 0.0015 0.0028 0.0034 0.005 Iphigenia 0.6 1.2 1.45 2.87 3.59 9.97 7.27

5

Conclusion

In this paper, we have defined an algorithm for computing persistent homology of a given filtration defined on a 2D mesh. Persistent homology is computed on different filtrations depending on a parameter δ. When δ = 0, the filtration coincides with the lower start filtration. When δ > 0 the filtration takes, pro-portionally to δ, more faces in each level. Our method provides high flexibility which allows easily to change the filtration due to the new parameter δ, allowing to speed-up (increasing δ) and giving to users a parameter allowing to tune their results depending on their needs.

One of our future work is to test the different possibilities for clusters re-garding to the parameter δ. For example, as one reviewer suggested, it would be interesting not only to take into account the distance between faces but also to consider the height of a face relatively to the seed before adding it to a cluster. Since we have observed in the experiments that our simplification filters small persistent homology events, we plan to provide theoretical results to this new approach stating that the filtration is stable with respect to δ. That is, the bottleneck distance between two filtrations of the same mesh is bounded by a function on δ. We think we can prove it using the classical result of Edelsbrunner et al on stability of persistence diagrams [4].

Finally, we plan to extend our work to non-orientable manifolds by using the generalized maps package (the non-orientable extension of combinatorial maps) of CGAL. We also would like to define a parallel version of our method: the combinatorial map simplification was already defined in parallel in [7] but we need now to study if it is possible to parallelize some parts of the AT-model computation algorithm. Extension in nD could be given based on the theoreti-cal results for removal and contraction operations in any dimension given in [16, 17]. Indeed, all basic tools used in this work, combinatorial maps, removal / con-traction operations and AT-model computation, are defined in any dimension. Acknowledgments. This research has been partially supported by MINECO, FEDER/UE under grant MTM2015-67072-P. We thank the anonymous review-ers for their valuable comments.

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